Preface

Least square method is the most commonly used data fitting method . This paper starts with the least square method Concept Starting with , The method of linear fitting is introduced Mechanism ; After through C Language Algorithm implementation ; Finally, it expounds the application of least square linear fitting Shortcomings and shortcomings .

One 、 Least square method

Least square method （ Also known as the least square method ） It's a mathematical optimization technique . It finds the best function matching of data by minimizing the square sum of errors .

With the simplest $y$ About $x$ The linear equation of ：$y={\beta }_0+{\beta }_{1}x$ For example . Suppose the coordinates of some given points are ：( $x_1,y_1$ ) , ( $x_2,y_2$) ⋯( $x_j,y_j$) ⋯( $x_n,y_n$); Define ordinates $y$ Residual of $Residual$ Is the difference between the estimated value and the observed value , The formula is as follows ：
$$Residual=y_i-({\beta }_0+{\beta }_{1}x)$$;

$${\beta }_0{=}_n^{\underline{\Sigma y_i}}{-}_n^{\underline{a\Sigma x_i}}$$

$${\beta }_1{=}_{n\Sigma {x_i}^2-(\Sigma x_i{)}^2}^{\underline{n\Sigma x_i{y}_i-\Sigma x_i\Sigma y_i}}$$
The purpose of the general least square method is to make the fitting error （ Residual sum ） Minimum , That is, the least square method is to find the parameters of a group of lines , Minimize the objective function .

Two 、 Code implementation (C Language )

The code is as follows ：

/**************************************************************/
//  Least square linear fitting (y = β0 + β1x)
//  notes ： This procedure only considers dependent variables y There are errors , The argument is given as X_sample
/**************************************************************/
float β1;
float β0;
const int16_t X_sample[100] =
	{
    1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,
	21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,
	41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,
	61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,
	81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100};

//  Sum of squares 
float Square_sum(int16_t* data, uint16_t len)
{
    
	uint16_t i;
	float z = 0;
	for (i = 0; i < len; i++)
	{
    
		z = z + (*data)*(*data);
		data++;
	}
	return z;
}

//  Vector product 
float X_By_Y(int16_t* X, int16_t* Y, uint16_t len)
{
    
	uint16_t i;
	float z = 0;
	for (i = 0; i < len; i++)
	{
    
		z = z + X[i] * Y[i];
	}
	return z;
}

//  Least square linear fitting 
float Line_fit(int16_t *data, uint16_t len)
{
    
	float X_square     = Square_sum((int16_t*)X_sample, len);
	float X_multiply_Y = X_By_Y((int16_t*)X_sample, data, len);

	float X_sum = SFLeaSum16s((int16_t*)X_sample, len);
	float Y_sum = SFLeaSum16s(data, len);

	β0= (X_square*Y_sum - X_sum*X_multiply_Y)/(len*X_square - X_sum*X_sum);
	β1= (len*X_multiply_Y - Y_sum*X_sum)/(len*X_square - X_sum*X_sum);

	return β1;
}

float Line_return(int16_t X)
{
    
	return β0 + β1*X;
}

3、 ... and 、 Defects and deficiencies

1. Sensitive to outliers

Observe the result of its algebraic solution , The least square method only uses the mean and variance information of points , Therefore, it is only applicable to the case of ordinary noise , When there are outliers , The general least square method is very sensitive to outliers , May change the final result , It needs to be solved by other methods .

2. The error of independent variables is not considered

Only when there is no deviation in the independent variable , Or the deviation of the independent variable can be ignored within a certain range without timing , The least square method is more applicable . Because the general least square method only considers the dependent variable $y$ There are errors , Independent variables are not considered $x$ The error of the , Therefore, its application conditions have certain limitations .

3. There are unsolvable cases

When the line to be fitted is perpendicular or nearly perpendicular to $x$ Axial time , The slope is infinite , The least square method is not applicable .

summary

1） Least square method is the most commonly used data fitting method , By minimizing the sum of squares of errors, the optimal function matching of the data is found ;
2） The least square method only uses the mean and variance information of points , Sensitive to outliers ;
3） The least square method only considers the dependent variable $y$ There are errors , But when the horizontal and vertical coordinates of the point have errors , And can not be ignored , The general least square method is not applicable ;
4） The least square method cannot be solved （ The fitted line is perpendicular or nearly perpendicular to $x$ Axis ）.

The above is the whole content of this article , I hope this paper can help you understand and use the least square method for linear fitting .
Of course , If there is any mistake or imprecision in the content of this article , Please also point out in time , thank you ！

Reference resources

Linear regression —— Least square fitting - David Xu 2014 - Blog Garden
Linear fitting 1- Least square method _ningzian The blog of -CSDN Blog